Problem 18 - Advanced Engineering Mathematics

Question

Find all values of \(z\) satisfying the given equation. \(\sinh z=-1\)

Step-by-Step Solution

Answer

The solution is \( z = \ln(-1 + \sqrt{2}) + ni\pi \), where \( n \) is an integer.

Step 1: Understand the Hyperbolic Sine Function

The hyperbolic sine function, denoted as \( \sinh z \), is defined as \( \sinh z = \frac{e^z - e^{-z}}{2} \). It is similar to the sine function in trigonometry but is based on the exponential function.

Step 2: Set Up the Equation

We start with the given equation: \( \sinh z = -1 \). Substitute the definition of \( \sinh z \) into this equation to get:\[ \frac{e^z - e^{-z}}{2} = -1 \]

Step 3: Solve for the Exponential Terms

Multiply both sides by 2 to eliminate the fraction, resulting in:\[ e^z - e^{-z} = -2 \] Change \( e^{-z} \) to \( \frac{1}{e^z} \):\[ e^z - \frac{1}{e^z} = -2 \].

Step 4: Substitute \( e^z = x \)

Let \( x = e^z \). Then the equation becomes:\[ x - \frac{1}{x} = -2 \] Multiply through by \( x \) to clear the fraction:\[ x^2 - 1 = -2x \].

Step 5: Rearrange to Form a Quadratic Equation

Bring all terms to one side of the equation to form a quadratic equation:\[ x^2 + 2x - 1 = 0 \].

Step 6: Solve the Quadratic Equation

Apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 2, \text{ and } c = -1 \):\[ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{-2 \pm \sqrt{8}}{2} = \frac{-2 \pm 2\sqrt{2}}{2} \]. This gives solutions: \[ x = -1 + \sqrt{2} \] and \[ x = -1 - \sqrt{2} \].

Step 7: Solve for \( z \)

Recall \( x = e^z \). Since \( x = -1 - \sqrt{2} \) is negative and can't be the exponential function result for any real \( z \), discard it. Solve \( e^z = -1 + \sqrt{2} \):\[ z = \ln(-1 + \sqrt{2}) + ni\pi \] (The solution family due to periodicity, \( n \) is an integer for periodicity, accounting for all complex solutions.)

Step 8: Verify the Principal Solution

Verify by substitution if \( z = \ln(-1 + \sqrt{2}) + ni\pi \) satisfies the original equation. Indeed, substituting \( z = \ln(-1 + \sqrt{2}) \) back gives \( \sinh(z) = -1 \). This also aligns with properties of hyperbolic functions.

Key Concepts

Hyperbolic Sine Function

The hyperbolic sine function, represented as \( \sinh z \), is a function that mirrors the behavior of the traditional sine function, but it is framed in terms of exponential functions rather than trigonometric ones. This function is defined by the equation:

• \( \sinh z = \frac{e^z - e^{-z}}{2} \)

Instead of oscillating like its trigonometric counterpart, the hyperbolic sine exhibits exponential growth. Understanding this definition is crucial because it transforms problems involving \( \sinh z \) into those involving exponential expressions. This shift allows us to use algebraic techniques like solving equations more directly. Whenever you're dealing with \( \sinh \), keep in mind that it relates closely to the exponential functions \( e^z \) and \( e^{-z} \).

Exponential Functions

Exponential functions form the backbone of the hyperbolic sine function and are vital in various mathematical problems. An exponential function is typically of the form \( e^x \), where \( e \) is the base of natural logarithms, approximately equal to 2.718. Exponential functions are known for modeling growth and decay because they increase rapidly.

• They have the form \( e^z \), where \( z \) is any complex or real number.
• In the context of complex numbers, they're used to describe rotations and oscillations.

In our case, converting \( e^z \) into \( x = e^z \) simplifies the solving process of equations involving \( \sinh \). Understanding how exponential functions work will aid in visualizing solutions that seem complex at first glance.

Quadratic Equations

A quadratic equation is a polynomial equation of degree two and takes the form:

• \( ax^2 + bx + c = 0 \)

Quadratic equations play a pivotal role in this specific problem because we transformed our hyperbolic sine equation into a quadratic format. Solving quadratic equations often involves using the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This method allows us to find solutions for \( x \) in the given quadratic. Notably, quadratics can have real or complex roots. In our example, knowing how to rearrange terms to form the quadratic opened the pathway to find solutions efficiently.

Complex Numbers

Complex numbers form a critical element when solving equations that involve periodicity and trigonometric or hyperbolic functions. A complex number is of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \).

• In many scenarios, like our equation \( \sinh z = -1 \), solutions naturally extend into the complex plane.
• The formula \( z = \ln(x) + ni\pi \), where \( n \) is an integer, showcases how periodicity in complex numbers is addressed.

The imaginary part \( ni\pi \) is crucial because it reflects the repeating nature of exponential functions involving complex arguments. Understanding complex numbers allows us to confidently tackle equations that extend beyond real-number solutions.